Find the Maclaurin polynomials of orders n 0 1 2 3 and 4 an

Find the Maclaurin polynomials of orders n = 0, 1, 2, 3, and 4, and then find the nth Maclaurin polynomials for the function in sigma notation. E^-2x Use sigma notation to write the Taylor series about x = x_0 for the function. 1/x + 2_0=3 lnx; x_0 = 1

Solution

answer-1

Maclaurin serie centered around a=0.

So what you have now its the following:

f(x) =f(0)+f\'(0)+(f\"(0)/2)x^2+(f\"\'(0)/3!)x^3+...

f(0) = e^-2(0) = 1
first derivative of e^-2x =   -2(e^-2x) (use chain rule)
f\'(0) = -2
second derivative is 4(e^-2x)
f\'\'(0) = 4
third derivative = -8(e^-2x)
f\'\'\'(0) = - 8
fourth derivative = 16 (e^-2x)
f\'\'\'\'(0) = 16

1-2/1!x^1+4/2!x^2 - 8 /3!x^3 +16 /4!x^4+...

the more compact form would be
sum(n=0, infinity) (-1)^n(2n/n!)x^n